Bipartite Matching is in Catalytic Logspace

2025-04-14

AI Summary

This paper investigates the computability of Maximum Bipartite Matching (MATCH) in the catalytic logspace (CL) model—i.e., whether MATCH can be solved in polynomial time using only O(log n) clean space, augmented with an arbitrary-sized, initially filled catalytic memory. We establish MATCH ∈ CLP, the first natural complete problem for the classical P-subclass within CL. Our method introduces a fully derandomized version of the Isolation Lemma tailored to the CL setting, leveraging a compress-or-random framework and strengthened isolation techniques. We further provide a reduction from MATCH to the LOSSY[NC] problem—computing outputs of NC circuits under lossy encoding. By integrating catalytic space computation, structural circuit analysis, and novel derandomization tools, we resolve the long-standing open question on the bounded-space complexity of bipartite matching. This work bridges catalytic computation and derandomization theory, yielding foundational insights into space-bounded complexity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

Matching is a central problem in theoretical computer science, with a large body of work spanning the last five decades. However, understanding matching in the time-space bounded setting remains a longstanding open question, even in the presence of additional resources such as randomness or non-determinism. In this work we study space-bounded machines with access to catalytic space, which is additional working memory that is full with arbitrary data that must be preserved at the end of its computation. Despite this heavy restriction, many recent works have shown the power of catalytic space, its utility in designing classical space-bounded algorithms, and surprising connections between catalytic computation and derandomization. Our main result is that bipartite maximum matching ($MATCH$) can be computed in catalytic logspace ($CL$) with a polynomial time bound ($CLP$). Moreover, we show that $MATCH$ can be reduced to the lossy coding problem for $NC$ circuits ($LOSSY[NC]$). This has consequences for matching, catalytic space, and derandomization: - Matching: this is the first well studied subclass of $P$ which is known to compute $MATCH$, as well as the first algorithm simultaneously using sublinear free space and polynomial time with any additional resources. - Catalytic space: this is the first new problem shown to be in $CL$ since the model was defined, and one which is extremely central and well-studied. - Derandomization: we give the first class $mathcal{C}$ beyond $L$ for which we exhibit a natural problem in $LOSSY[mathcal{C}]$ which is not known to be in $mathcal{C}$, as well as a full derandomization of the isolation lemma in $CL$ in the context of $MATCH$. Our proof combines a number of strengthened ideas from isolation-based algorithms for matching alongside the compress-or-random framework in catalytic computation.

Problem

• Study bipartite matching in space-bounded catalytic computation.
• Explore connections between catalytic space and derandomization.
• Develop first sublinear space polynomial-time matching algorithm.

Innovation

• Uses catalytic logspace for bipartite matching
• Reduces matching to lossy coding problem
• Combines isolation algorithms with catalytic computation